Question

1. sara wants to save $5,000 to put towards college. she has a savings account with $320. each month, she deposits $180 from her part - time job in the savings account.

a. write an equation that represents her savings account balance, y, after x months.
b. how much does she have after 3 months?
c. graph saras savings account balance and goal using desmos. use this graph to answer the following questions.
d. if she continues saving $180 per month, how long will it take her to reach her goal? how can you find that point on the graph?
e. approximately how many months earlier will sara meet her saving goal if she saves $240 per month, instead of $180 per month?
f. if sara wants to reach her $5000 goal in one year, approximately how much would she need to save each month?

Explanation:

Step1: Define linear equation structure

The balance follows $y = mx + b$, where $b$ is initial amount, $m$ is monthly deposit.

Step2: Plug in known values

Initial amount $b = 320$, monthly deposit $m = 180$.
Expression: $y = 180x + 320$

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Step1: Substitute $x=3$ into equation

Use the equation from part a.
Expression: $y = 180(3) + 320$

Step2: Calculate total balance

Compute the product then sum.
Expression: $y = 540 + 320 = 860$

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Step1: Set $y=5000$, solve for $x$

Use the equation $5000 = 180x + 320$.

Step2: Isolate the $x$-term

Subtract initial amount from both sides.
Expression: $180x = 5000 - 320 = 4680$

Step3: Solve for $x$

Divide by monthly deposit.
Expression: $x = \frac{4680}{180} = 26$
On the graph, this is the $x$-value where the line $y=180x+320$ intersects the horizontal line $y=5000$.

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Step1: Find time for $240/month$

Set $5000 = 240x + 320$.

Step2: Isolate $x$-term

Expression: $240x = 5000 - 320 = 4680$

Step3: Solve for new $x$

Expression: $x = \frac{4680}{240} = 19.5$

Step4: Calculate time difference

Subtract new time from original time.
Expression: $26 - 19.5 = 6.5$

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Step1: Set $x=12$, solve for $m$

Use $5000 = 12m + 320$.

Step2: Isolate the $m$-term

Expression: $12m = 5000 - 320 = 4680$

Step3: Solve for monthly deposit

Expression: $m = \frac{4680}{12} = 390$

Answer:

a. $y = 180x + 320$
b. $\$860$
c. (Graph: Plot the line $y=180x+320$ and horizontal line $y=5000$ on Desmos)
d. 26 months; find the intersection of $y=180x+320$ and $y=5000$ on the graph.
e. 6.5 months earlier
f. $\$390$ per month